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13 Let (X, d) be a complete metric space and / : X —)• R be a proper lsc and lower bounded function. Suppose that for every x G X such that f(x) > inf / , there exists x G X\{x} such that f(x) + d(x,x) < f(x). Prove that argmin / ^ 0 and d(x, argmin / ) < f(x) — inf / for every x E X. 14 Let (X, d) be a complete metric space, / : X —• R be a proper lsc lower bounded function and 3£ : X =t X. Assume that for every x G X there exists y G 3l(x) such that d(x,y) + f(y) < f(x). e. 15 Let (X, ||||) be a normed space and / : X -> R.

One follows the same argument as in the proof of the preceding lemma, but with Un := pUx and Vn := nBy for n > 1. Consider y' 6 nBy and take p, £]0,1[ such that y :— (1 — p) y' £ c\"R(pUx)- We find the sequence {{xn,yn))n>l C grft such that (xn) C pUx and v„ := y - 2/1 - py2 ^ " _ 1 2 / n £ pnr)BY for n > 1. _1 Hence (u n ) -> 0 and /x" 2/„ = u„_i - vn, whence \\yn\\ < 77(1 + p) for n > 1. Taking An := (1 — p)pn^1 > 0 for n > 1, X) n >i ^" = 1> * n e s e r i e s S « > i ^n%n is Cauchy and the series X3n>i ^n2/n is convergent with sum y'.

2(h), ]JieI Bt is an ideally convex subset of r i i e / ( ^ j x ^ ) - The space X := riig/ ^i 1S a Frechet space; let Y := FJie/ ^i- Consider the set B:= {((xi)ieI,(yi)ieI) £XxY\ (xi,yi) £ Bi Vt 6 / } . Since T : I l i e / ( X * x y4) - • X x y , T ( ( x i , y i ) i 6 / ) := ((xi)ieI,(yi)iei) is an isomorphism of topological vector spaces, B — T (Y\ieI Bi) is ideally convex. As C := riig/ C* = ^VY(B), C is li-convex. D Before stating other properties of li-convex and lcs-closed sets, let us define some notions and notations related to multifunctions.

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Approximation Theory and Functional Analysis by J. Prolla

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